NCERT Mathematics · Class 10 · English · 2026-27
Surface Areas and Volumes — Exercise 12.2
Independently written, step-by-step solutions. Prompts below are concise paraphrases; use the official textbook for the complete question wording.
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Question 1
A radius-1 cm hemisphere carries a cone of radius and height 1 cm. Find volume in terms of π.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Cone volume=(1/3)πr²h=π/3. Hemisphere volume=(2/3)πr³=2π/3. Their interiors do not overlap, so add.
Resource ID: CB-10-M-CH12-EX12.2-Q1
Question 2
A diameter-3 cm model is 12 cm long, with a cone of height 2 cm at each end of its cylinder. Find internal volume.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Radius=1.5 cm; cylinder length=12−2−2=8 cm. Volume=π(1.5)²8+2[(1/3)π(1.5)²2]=18π+3π=21π.
Resource ID: CB-10-M-CH12-EX12.2-Q2
Question 3
Estimate syrup in 45 gulab jamuns, each length 5 cm and diameter 2.8 cm, if syrup occupies 30% of each.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Each has r=1.4 and cylinder length 5−2r=2.2 cm. Volume per jamun=πr²h+(4/3)πr³=(22/7)(1.4)²(2.2)+(4/3)(22/7)(1.4)³≈25.05067 cm³. Syrup=0.30×45×that volume.
Resource ID: CB-10-M-CH12-EX12.2-Q3
Question 4
A 15×10×3.5 cm wooden pen stand has four conical holes of radius 0.5 cm and depth 1.4 cm. Find wood volume.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Cuboid volume=525 cm³. Total removed volume=4[(1/3)(22/7)(0.5)²(1.4)]=22/15 cm³. Subtract this from 525.
Resource ID: CB-10-M-CH12-EX12.2-Q4
Question 5
A full cone of height 8 cm and radius 5 cm overflows by a quarter of its capacity when radius-0.5 cm spheres are submerged. Find count.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Cone capacity=(1/3)π×25×8=200π/3. Overflow=(1/4)capacity=50π/3. One sphere displaces (4/3)π(0.5)³=π/6. With complete submersion, n(π/6)=50π/3.
Resource ID: CB-10-M-CH12-EX12.2-Q5
Question 6
Find the mass of an iron pole with cylinders (radius 12 cm,height 220 cm) and (radius 8 cm,height 60 cm), density 8 g/cm³. Use π=3.14.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Total volume=3.14[12²×220+8²×60]=3.14(31680+3840)=111532.8 cm³. Mass=volume×8=892262.4 g. Divide by 1000 for kilograms.
Resource ID: CB-10-M-CH12-EX12.2-Q6
Question 7
A cone (r=60,h=120 cm) on a hemisphere (r=60 cm) is submerged in a full cylinder (r=60,h=180 cm). Find water left.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Cylinder capacity=π60²180=648000π cm³. Solid volume=(1/3)π60²120+(2/3)π60³=144000π+144000π=288000π. It fits inside and touches the bottom. Remaining water=capacity−displacement=360000π.
Resource ID: CB-10-M-CH12-EX12.2-Q7
Question 8
A spherical vessel has a cylindrical neck 8 cm long, diameter 2 cm; sphere diameter 8.5 cm. Is a measured capacity of 345 cm³ correct? Use π=3.14.
Step-by-step solution
Use π=22/7 unless the question specifies another value. Count only exposed surfaces for surface area; add or subtract solid volumes for capacity.
Under the textbook’s combination model, sphere radius=4.25 cm and neck radius=1 cm. Volume=(4/3)3.14(4.25)³+3.14(1)²8≈321.39208+25.12=346.51208 cm³. Compare with the stated 345 cm³.
Resource ID: CB-10-M-CH12-EX12.2-Q8